A simple range characterization for spherical mean transform in odd dimensions and its applications

Technical Summary: A Simple Range Characterization for Spherical Mean Transform in Odd Dimensions and Its Applications

Problem Statement
The paper addresses the problem of characterizing the range of the Spherical Mean Transform (SMT), also known as the spherical Radon transform, for functions supported in the unit ball $B \subset \mathbb{R}^n$ where $n$ is an odd integer ($n \ge 3$). Specifically, the transform maps a function $f$ to its averages over spheres centered on the boundary of the unit ball ($S^{n-1}$) with varying radii $t \in (0, \infty)$. While the injectivity of this transform and various inversion formulas are well-established, existing range characterizations (conditions that a function must satisfy to be in the image of the transform) are often prohibitively complex for constructive proofs. The authors aim to derive a simpler, more tractable set of range conditions to facilitate the construction of functions with specific support properties and to investigate the Unique Continuation Property (UCP) for the SMT.

Methodology
The authors employ a combination of harmonic analysis, special function theory, and differential operator techniques:

1. Spherical Harmonics Expansion: The problem is decomposed by expanding both the source function $f$ and the transform data $g = Rf$ into spherical harmonics. This reduces the general problem to analyzing radial functions for each harmonic degree $m$.
2. Differential Operators and Symmetry: For radial functions, the authors derive a necessary condition involving a linear differential operator $L_k$ of order $k = (n-3)/2$. They establish that the transform of a radial function satisfies a specific symmetry relation between values at $1-t$ and $1+t$ (or equivalently $t$ and $2-t$) after applying a weighting factor $t^{n-2}$.
3. Combinatorial Analysis: The proof of the general range condition relies on intricate combinatorial identities. The authors utilize the Egorychev technique (contour integrals for binomial coefficients) and a specialized version of the Faà di Bruno formula for the operator $D = \frac{1}{t}\frac{d}{dt}$ to handle higher-order derivatives of composite functions arising in the integral kernels.
4. Special Functions: A central technical component involves the derivation of a "cross product" identity for spherical Bessel functions of the first kind ($j_\alpha$) and second kind ($y_\alpha$). This identity relates the Hankel transforms of functions satisfying the derived symmetry conditions.
5. Ordinary Differential Equations (ODEs): To prove the existence of counterexamples to the UCP, the authors construct non-trivial smooth functions with compact support that satisfy the derived range conditions. This is achieved by solving linear ODEs with variable coefficients and carefully choosing boundary conditions to ensure the function and its derivatives vanish at specific points.

Key Results

1. Range Characterization for Radial Functions (Theorem 1.1):
   A function $g \in C^\infty_c((0, 2))$ is the SMT of a radial function $f \in C^\infty_c(B)$ if and only if $h(t) = t^{n-2}g(t)$ satisfies the symmetry relation:
   $$[L_k h](1-t) = [L_k h](1+t) \quad \text{for all } t \in [0, 1],$$
   where $L_k$ is a specific linear differential operator of order $k = (n-3)/2$. This condition is significantly simpler than previous characterizations.

2. Range Characterization for General Functions (Theorem 1.4):
   For a general function $g \in C^\infty_c(S^{n-1} \times (0, 2))$, let $g_{m,l}(t)$ be its spherical harmonic coefficients. Then $g = Rf$ if and only if for each $(m, l)$, the function $h_{m,l}(t) = t^{n-2}g_{m,l}(t)$ can be written as $h_{m,l} = D^m \phi_{m,l}$ for some $\phi_{m,l} \in C^\infty_c((0, 2))$, where $\phi_{m,l}$ satisfies the symmetry relation:
   $$[L_{m+k} \phi_{m,l}](1-t) = [L_{m+k} \phi_{m,l}](1+t).$$

3. Cross Product Identity for Spherical Bessel Functions (Theorem 1.8):
   The proof yields a remarkable identity for any $h \in C^\infty_c((0, 2))$ satisfying the range symmetry condition:
   $$\left( \int_0^\infty j_{k+1/2}(\lambda t) t h(t) dt \right) y_{k+1/2}(\lambda) = \left( \int_0^\infty y_{k+1/2}(\lambda t) t h(t) dt \right) j_{k+1/2}(\lambda).$$
   The authors note this identity provides an infinite family of cross-product relations for spherical Bessel functions not previously found in literature.

4. Counterexamples to Unique Continuation (Theorem 1.6 and Corollary 1.7):
   Using the range characterization, the authors construct explicit non-trivial functions $f \in C^\infty_c(B)$ that vanish on an open set $U \subset B$ (specifically a ball around the origin or any open subset) while their SMT vanishes on all spheres intersecting $U$. This proves that the Unique Continuation Property (UCP) does not hold for the SMT in odd dimensions.

Significance and Claims
The paper claims that its primary contribution is the derivation of a simple and constructive range characterization for the SMT in odd dimensions. Unlike previous complex conditions, these new symmetry relations involving differential operators allow for the explicit construction of functions in the range with specific support constraints.

The authors highlight two main applications of this result:

• Constructive Proofs: The simplicity of the conditions enables the construction of counterexamples to the UCP, a property that holds for many other integral transforms (like the X-ray transform in odd dimensions) but fails here due to the local nature of the SMT inversion in odd dimensions.
• Special Functions Theory: The derived cross-product identity for spherical Bessel functions is presented as a result of independent interest in the theory of special functions, illuminating the structure of zeros of the Hankel transform for functions in the range of the SMT.

The authors explicitly state that while the UCP fails in odd dimensions, whether it holds for the SMT in even dimensions remains an open question. They also note that their work complements a separate study (referenced as [8]) which provides a range characterization for even dimensions based on integral operators rather than differential ones.